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Theorem sotri2 5956

Description: A transitivity relation. (Read 𝐴 ≤ 𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)

**Hypotheses**
Ref	Expression
soi.1	⊢ 𝑅 Or 𝑆
soi.2	⊢ 𝑅 ⊆ (𝑆 × 𝑆)

**Assertion**
Ref	Expression
sotri2	⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

**Proof of Theorem sotri2**
Step	Hyp	Ref	Expression
1		soi.2	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
2	1	brel 5581	. . . 4 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
3	2	simpld 498	. . 3 ⊢ (𝐵𝑅𝐶 → 𝐵 ∈ 𝑆)
4		soi.1	. . . . . . 7 ⊢ 𝑅 Or 𝑆
5		sotric 5465	. . . . . . 7 ⊢ ((𝑅 Or 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴𝑅𝐵)))
6	4, 5	mpan 689	. . . . . 6 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴𝑅𝐵)))
7	6	con2bid 358	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐵 = 𝐴 ∨ 𝐴𝑅𝐵) ↔ ¬ 𝐵𝑅𝐴))
8		breq1 5033	. . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐵𝑅𝐶 ↔ 𝐴𝑅𝐶))
9	8	biimpd 232	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
10	4, 1	sotri 5954	. . . . . . 7 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
11	10	ex 416	. . . . . 6 ⊢ (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
12	9, 11	jaoi 854	. . . . 5 ⊢ ((𝐵 = 𝐴 ∨ 𝐴𝑅𝐵) → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
13	7, 12	syl6bir 257	. . . 4 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶)))
14	13	com3r 87	. . 3 ⊢ (𝐵𝑅𝐶 → ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶)))
15	3, 14	mpand 694	. 2 ⊢ (𝐵𝑅𝐶 → (𝐴 ∈ 𝑆 → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶)))
16	15	3imp231 1110	1 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 Or wor 5437 × cxp 5517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-po 5438 df-so 5439 df-xp 5525

This theorem is referenced by: supsrlem 10522

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